Families of linear-quadratic problems: Continuity properties

In this paper we investigate for given one-parameter families of linear time-invariant finite-dimensional systems the parameter dependence of the linear-quadratic optimal cost, optimal control inputs, optimal state-trajectories, and optimal outputs. It is shown that results that have been obtained in the past in the context of the problem of cheap control can in fact be generalized to a much broader class of parameter-dependent cost-functionals, including cost functionals in which for every parameter value the weighting matrix of the control inputs is singular. Essentially, only two assumptions on the parameter dependence of the cost-functionals are required in order to have continuity of the optimal cost and optimal control inputs with respect to the underlying parameter. One assumption is concerned with the continuity of the weighting matrices with respect to this parameter, the other with the monotonicy of the weighting matrices with respect to the parameter. Instrumental in our development is a characterization of the linear-quadratic optimal cost in terms of the so-called dissipation inequality. The results obtained are applied to the problem of cheap control and to a problem of priority control. The latter provides an example of a family of quadratic cosl-functionals with a polynomial parameter dependence.     

Equivalence of rational representations of behaviors

This article deals with the equivalence of representations of behaviors of linear differential systems. In general, the behavior of a given linear differential system has many different representations. In this paper we restrict ourselves to kernel and image representations. Two kernel representations are called equivalent if they represent one and the same behavior. For kernel representations defined by polynomial matrices, necessary and sufficient conditions for equivalence are well known. In this paper, we deal with the equivalence of rational representations, i. e. kernel and image representations that are defined in terms of rational matrices. As the first main result of this paper, we will derive a new condition for the equivalence of rational kernel representations of possibly noncontrollable behaviors. Secondly we will derive conditions for the equivalence of rational representations of a given behavior in terms of the polynomial modules generated by the rows of the rational matrices. We will also establish conditions for the equivalence of rational image representations. Finally, we will derive conditions under which a given rational kernel representation is equivalent to a given rational image representation.     

Tracking and regulation in the behavioral framework

Given a plant together with an exosystem generating the disturbances and the reference signals, the problem of asymptotic tracking and regulation is to find a controller such that the plant variable tracks the reference signal regardless of the disturbance acting on the system. If a controller achieves this design objective, we call it a regulator for the plant with respect to the given exosystem. In this paper, we formulate the asymptotic tracking and regulation problem in the behavioral framework, with control by interconnection.     

Polynomial Embedding Algorithms for Controllers in a Behavioral Framework

In this correspondence, we will establish polynomial algorithms for computation of controllers in the behavioral approach to control, in particular for the computation of controllers that regularly implement a given desired behavior and for controllers that achieve pole placement and stabilization by behavioral full interconnection and partial interconnection. These synthesis problems were studied before in articles by Belur and Trentelman, Rocha and Wood, and Willems in the reference section. In the algorithms, we will apply ideas around the unimodular and stable embedding problems. The algorithms that are presented in this correspondence can be implemented by means of the Polynomial Toolbox of Matlab.     

Reverse Engineering Ancient Greek Ceramics: Morphological and Spectral Characterization of Replicates

Athenian pottery, the ceramics produced in the Attica region of Greece between the 6th and 4th centuries B.C., is considered a benchmark technological achievement of the preindustrial world. This work advances our understanding of the firing protocols employed by the ancient Greeks to produce their black‐on‐red designs by characterizing replicates painted with a refined illite clay and fired under oxidizing/reducing/reoxidizing conditions (three‐stage firing). Systematically varying the temperature, atmosphere, and duration of each firing stage within the three‐stage firing scheme allowed the conditions necessary to obtain black and red gloss, both of which are observed on ancient vessels, to be determined. The morphology and elemental distribution of particles formed within the gloss thus formed were characterized using transmission electron microscopy and spectrocolorimetric measurements. Comparison of the results obtained from ancient sherds with those obtained from the replicate samples provides a means of estimating the firing conditions used to create the ancient vessels.     

On the regular implementability of nD systems

In this short paper we study the problem of finding necessary and sufficient conditions for regular implementability by partial interconnection for nD system behaviors. In [M.N. Belur, H.L. Trentelman, Stabilization, pole placement and regular implementability, IEEE Trans. Automat. Control 47(5) (2002) 735–744.] such conditions were obtained in the context of 1D systems. In the present paper we show that the conditions obtained in [M.N. Belur, H.L. Trentelman, Stabilization, pole placement and regular implementability, IEEE Trans. Automat. Control 47(5) (2002) 735–744.] are no longer valid in general in the nD context. We also show that under additional assumptions, the conditions still remain relevant. We also reinvestigate the conditions for regular implementability by partial interconnection in terms of the canonical controller that were obtained in [P. Rocha, Canonical controllers and regular implementation of nD behaviors, Proceedings of the 16th IFAC World Congress, Prague, Czech Republic, 2005.]. Using the geometry of the underlying modules we generalize a result on regular implementability from the 1D to the nD case. Finally, we study how, in the 1D context, the conditions from [M.N. Belur, H.L. Trentelman, Stabilization, pole placement and regular implementability, IEEE Trans. Automat. Control 47(5) (2002) 735–744; P. Rocha, Canonical controllers and regular implementation of nD behaviors, Proceedings of the 16th IFAC World Congress, Prague, Czech Republic, 2005.] are connected.     

Synthesis of dissipative systems using quadratic differentialforms: Part I

The problem discussed is that of designing a controller for a linear system that renders a quadratic functional nonnegative. Our formulation and solution of this problem is completely representation-free. The system dynamics are specified by a differential behavior, and the performance is specified through a quadratic differential form. We view control as interconnection: a controller constrains a distinguished set of system variables, the control variables. The resulting behavior of the to-be-controlled variables is called the controlled behavior. The constraint that the controller acts through the control variables only can be succinctly expressed by requiring that the controlled behavior should be wedged in between the hidden behavior, obtained by setting the control variables equal to zero, and the plant behavior, obtained by leaving the control variables unconstrained. The main result is a set of necessary and sufficient conditions for the existence of a controlled behavior that meets the performance specifications. The essential requirement is a coupling condition, an inequality that combines the storage functions of the hidden behavior and the orthogonal complement of the plant behavior     

On flat systems behaviors and observable image representations

In this short paper we propose a definition of flatness for systems not necessarily given in input/state/output representation. A flat system is a system for which there exists a mapping such that the manifest system behavior is equal to the image of this mapping, and such that the latent variable appearing in this image representation can be written as a function of the manifest variable and its derivatives up to some order. For linear differential systems, flatness is equivalent to controllability. We will generalize the main theorem of Levine and Nguyen (Systems Control Lett. 48 (2003) 69) to general linear differential systems.     

Optimization of a chirped-pulse amplification Nd:glass laser

To allow us to achieve the highest focused intensity from a chirped-pulse amplification Nd:glass laser, a number of features of the system have been considered and optimized. These include the compressor geometry, the system aberrations, and the use of mixed glasses in the amplifier chain. Calculations for the laser with a single- or double-pass pulse compressor with 450-mm gratings are presented. These indicate that, for single pass, a reduction in pulse duration from 380 to 237 fs is possible when a phosphate is changed to a mixed phosphate-silicate glass system, and there is a corresponding increase of 44% in peak intensity at beam focus.     

Dissipativity preserving model reduction by retention of trajectories of minimal dissipation

We present a method for model reduction based on ideas from the behavioral theory of dissipative systems, in which the reduced order model is required to reproduce a subset of the set of trajectories of minimal dissipation of the original system. The passivity-preserving model reduction method of Antoulas (Syst Control Lett 54:361–374, 2005) and Sorensen (Syst Control Lett 54:347–360, 2005) is shown to be a particular case of this more general class of model reduction procedures.